arXiv subjects

Philippe G. LeFloch

Publications and source records attributed to Philippe G. LeFloch.

At least 145 records · Page 8Linked to original sources

arXiv2007

The minimum entropy principle for fluid flows in a nozzle with discontinuous cross-section

We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch, and Murat. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class of numerical approximations for this system.

math.NA↗

arXiv2007

The Riemann problem for the shallow water equations with discontinuous topography

We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the existence of two-parameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initial-value problem.

math.AP↗

arXiv2007

Why many theories of shock waves are necessary. Kinetic functions, equivalent equations, and fourth-order models

We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit under-compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by {\sl high-order} finite difference schemes, and uncover several new features of the kinetic function associated with with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model --especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa-Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points.

math.NA↗

arXiv2007

Convergent and conservative schemes for nonclassical solutions based on kinetic relations

We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions.

math.NA↗

arXiv2007

Finite volume schemes on Lorentzian manifolds

We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry accurately and generalize techniques and estimates that were known in the (flat) Euclidian setting, only. The strong convergence of the scheme then is then a consequence of the well-posed theory recently developed by Ben-Artzi and LeFloch for conservation laws on manifolds.

math.NA↗

arXiv2007

Definition and stability of Lorentzian manifolds with distributional curvature

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we investigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curvature tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.

gr-qc↗

arXiv2007

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.

math.DG↗

arXiv2007

Zero diffusion-dispersion limits for scalar conservation laws

We consider solutions of hyperbolic conservation laws regularized with vanishing diffusion and dispersion terms. Following a pioneering work by Schonbek, we establish the convergence of the regularized solutions toward discontinuous solutions of the hyperbolic conservation law. The proof relies on the method of compensated compactness in the $L^2$ setting. Our result improves upon Schonbek's earlier results and provides an optimal condition on the balance between the relative sizes of the diffusion and the dispersion parameters. A convergence result is also established for multi-dimensional conservation laws by relying on DiPerna's uniqueness theorem for entropy measure-valued solutions.

math.AP↗

arXiv2007

Hyperbolic conservation laws and spacetimes with limited regularity

Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows and general relativity. Recent work by the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either be fixed (shallow water equations on the sphere, for instance) or be one of the unknowns of the theory (Einstein-Euler equations of general relativity).

math.AP↗

arXiv2007

Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions

Solutions to conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme being second order accurate away from sonic points and extrema.

math.NA↗

arXiv2007

Conservation laws with vanishing nonlinear diffusion and dispersion

We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosity approximation introduced by Von Neumann in the 50's.

math.AP↗

arXiv2007

Boundary layers in weak solutions to hyperbolic conservation laws

This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing viscosity method and finite difference schemes (Lax-Friedrichs type schemes, Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L\infty bounds on the approximate solutions and so dealing with L\infty solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in ``Boundary conditions for nonlinear hyperbolic systems of conservation laws'', J. Diff. Equa. 71 (1988), 93--122. The local structure of those sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.

math.AP↗

arXiv2007

Propagating phase boundaries: formulation of the problem and existence via the Glimm scheme

In this paper we consider the hyperbolic-elliptic system of two conservation laws that describes the dynamics of an elastic material governed by a non-monotone strain-stress function. Following Abeyaratne and Knowles, we propose a notion of admissible weak solution for this system in the class of functions with bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary), and an nucleation criterion (for the appearance of new phase boundaries). We prove the L1 continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proof is based on Glimm's scheme and in particular on techniques going back to Glimm-Lax. In order to deal with the kinetic relation, we prove a result of pointwise convergence for the phase boundary.

math.AP↗

arXiv2007

Symmetries and global solvability of the isothermal gas dynamics equations

We study the Cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics, in which the pressure and the density are related by the $γ$-law equation $p(ρ) \sim ρ^γ$ with $γ=1$. Our results complete those obtained earlier for $γ>1$. We prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method. The proof relies on compensated compactness arguments and symmetry group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density. This property enables us to reduce our problem to that with a small initial density. One symmetry group associated with the linear hyperbolic equations describing all entropies of the Euler equations gives rise to a fundamental solution with initial data imposed to the line $ρ=1$. This is in contrast to the common approach (when $γ>1$) which prescribes initial data on the vacuum line $ρ=0$. The entropies we construct here are weak entropies, i.e. they vanish when the density vanishes. Another feature of our proof lies in the reduction theorem which makes use of the family of weak entropies to show that a Young measure must reduce to a Dirac mass. This step is based on new convergence results for regularized products of measures and functions of bounded variation.

math.AP↗

arXiv2007

A version of the Glimm method based on generalized Riemann problems

We introduce a generalization of Glimm's random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form building blocks in our scheme and allow us to take into account naturally the effects of the flux and source terms. To establish the nonlinear stability of these approximations, we investigate nonlinear interactions between generalized wave patterns. This analysis leads us to a global existence result for quasilinear hyperbolic systems with source-term, and applies, for instance, to the compressible Euler equations in general geometries and to hyperbolic systems posed on a Lorentzian manifold.

math.AP↗

arXiv2007

Entropy solutions of the Euler equations for isothermal relativistic fluids

We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the existence of globally defined, bounded measurable, entropy solutions with arbitrary large amplitude. An earlier result by Smoller and Temple for the same system covered solutions with bounded variation that avoid the vacuum state. The new framework proposed here provides entropy solutions in a larger function space and allows for the mass density to vanish and the velocity field to approach the speed of light. The relativistic Euler equations become strongly degenerate in both regimes, as the conservative or the flux variables vanish or blow-up. Our proof is based on the method of compensated compactness for nonlinear systems of conservation laws (Tartar, DiPerna) and takes advantage of a scaling invariance property of the isothermal fluid equations. We also rely on properties of the fundamental kernel that generates the mathematical entropy and entropy flux pairs. This kernel exhibits certain singularities on the boundary of its support and we are led to analyze certain nonconservative products (after Dal Maso, LeFloch, and Murat) consisting of functions of bounded variation by measures.

math.AP↗

arXiv2007

Nonlinear hyperbolic systems: Non-degenerate flux, inner speed variation, and graph solutions

We study the Cauchy problem for general, nonlinear, strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves, only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time-regularity of the graph solutions $(X,U)$ introduced by the second author, and propose a geometric version of our scheme; in turn, the spatial component $X$ of a graph solution can be chosen to be continuous in both time and space, while its component $U$ is continuous in space and has bounded variation in time.

math.AP↗

arXiv2006

Singular limits in phase dynamics with physical viscosity and capillarity

Following pioneering work by Fan and Slemrod who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid-vapor phase dynamics with {\sl physical} viscosity and capillarity effects taken into account. Following Dafermos we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis cover both the hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large Riemann data. The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic-elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity-viscosity ratio) are observed which prevent the solution to have globally bounded variation.

math.AP↗